A study in depth of f0(1370)

Claims have been made that f(0)(1370) does not exist. The five primary sets of data requiring its existence are refitted with suitable Breit-Wigner amplitudes. Major dispersive effects due to the opening of the 4 pi threshold are included for the first time; the sigma -> 4 pi amplitude plays a strong role. Crystal Barrel data on (p) over barp -> 3 pi(0) at rest require f(0)(1370) signals of at least 32 and 33 standard deviations (sigma) in S-1(0) and P-3(1) annihilation respectively. Furthermore, they agree within 5 MeV for mass and width. Data on (p) over barp ->eta eta pi(0) agree and require at least a 19 sigma contribution. This alone is sufficient to demonstrate the existence of f(0)(1370). BES II data for J/Psi ->phi pi(+)pi(-) contain a visible f(0)(1370) signal > 8 sigma. In all cases, a resonant phase variation is required. Cern-Munich data for pi pi elastic scattering are fitted well with the inclusion of some mixing between sigma, f(0)(1370) and f(0)(1500). Values of Gamma(2 pi) for f(2)(1565), rho(3)(1690), rho(3)(1990) and f(4)(2040) are determined.